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<div class="add">

<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="robbins1.html#L2" title="ROBBINS1:struct.2">ComplLattStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V4" title="LATTICES:attr.4">join-commutative</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V5" title="LATTICES:attr.5">join-associative</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="robbins1.html#V5" title="ROBBINS1:attr.5">Huntington</a> implies ( <font color="Maroon" title="c1">L</font> is <a href="robbins2.html#V2" title="ROBBINS2:attr.2">satisfying_MD_1</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="robbins2.html#V3" title="ROBBINS2:attr.3">satisfying_MD_2</a> ) )</span><br/>

<b>assume </b><a NAME="E1:66_2_1"/>
( <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V4" title="LATTICES:attr.4">join-commutative</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V5" title="LATTICES:attr.5">join-associative</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="robbins1.html#V5" title="ROBBINS1:attr.5">Huntington</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> is <a href="robbins2.html#V2" title="ROBBINS2:attr.2">satisfying_MD_1</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="robbins2.html#V3" title="ROBBINS2:attr.3">satisfying_MD_2</a> )</span><br/>

<b>then reconsider </b><font color="Maroon" title="c2">L'</font> = <font color="Maroon" title="c1">L</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="lattices.html#V4" title="LATTICES:attr.4">join-commutative</a> <a href="lattices.html#V5" title="LATTICES:attr.5">join-associative</a> <a href="robbins1.html#V5" title="ROBBINS1:attr.5">Huntington</a>  <a href="robbins1.html#L2" title="ROBBINS1:struct.2">ComplLattStr</a>  ;<br/>
<a NAME="E3:66_2_1"/><i><font color="Green" title="E60">A1</font></i>: 
<font color="Maroon" title="c2">L'</font> is <a href="robbins2.html#V2" title="ROBBINS2:attr.2">satisfying_MD_1</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L'</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="robbins2.html#D2" target="_self" title="ROBBINS2:def.2">ROBBINS2:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#NK4" title="ROBBINS1:NK.4">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#NK4" title="ROBBINS1:NK.4">+</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font></span><br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <font color="Maroon" title="c3">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <font color="Maroon" title="c3">x</font>
<b>by </b><i><a class="ref" href="robbins1.html#T3" title="ROBBINS1:th.3">ROBBINS1:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="robbins1.html#K6" title="ROBBINS1:func.6">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <font color="Maroon" title="c3">x</font>
<b>by </b><i><a class="ref" href="robbins1.html#D4" title="ROBBINS1:def.4">ROBBINS1:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c3">x</font>
<b>by </b><i><a class="ref" href="robbins1.html#T21" title="ROBBINS1:th.21">ROBBINS1:21</a></i>
;<br/>
<b>hence </b><a NAME="E2:66_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#NK4" title="ROBBINS1:NK.4">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#NK4" title="ROBBINS1:NK.4">+</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:66_2_1"/>
<font color="Maroon" title="c2">L'</font> is <a href="robbins2.html#V3" title="ROBBINS2:attr.3">satisfying_MD_2</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L'</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="robbins2.html#D3" target="_self" title="ROBBINS2:def.3">ROBBINS2:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#NK4" title="ROBBINS1:NK.4">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#NK4" title="ROBBINS1:NK.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="robbins1.html#NK4" title="ROBBINS1:NK.4">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> <a href="robbins1.html#NK4" title="ROBBINS1:NK.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="robbins1.html#NK4" title="ROBBINS1:NK.4">+</a> <font color="Maroon" title="c3">x</font></span>)</span></span><br/>

<b>set </b><font color="Maroon" title="c6">Z</font> = <font color="Maroon" title="c5">z</font> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <font color="Maroon" title="c4">y</font>;<br/>
<b>consider </b><font color="Maroon" title="c7">k</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L'</font><b> such that </b><br/><a NAME="E2:66_2_1_2"/><i><font color="Green" title="E61">A2</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c2">L'</font>
 <b>by </b><i><a class="ref" href="robbins1.html#D8" title="ROBBINS1:def.8">ROBBINS1:def 8</a></i>;<br/>
<i><font color="Green" title="E62">A3</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> = 
<font color="Maroon" title="c5">z</font> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D5" title="LATTICES:def.5">LATTICES:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c5">z</font> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <span class="p1">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c2">L'</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:66_2_1_2"><i><font color="Green" title="E61">A2</font></i></a>, <a class="ref" href="robbins1.html#T4" title="ROBBINS1:th.4">ROBBINS1:4</a></i>
<br/>.= 
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c2">L'</font>
<b>by </b><i><a class="ref" href="robbins1.html#T20" title="ROBBINS1:th.20">ROBBINS1:20</a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <font color="Maroon" title="c4">y</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <font color="Maroon" title="c4">y</font></span>)</span>
<b>by </b><i><a class="ref" href="robbins1.html#T3" title="ROBBINS1:th.3">ROBBINS1:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="robbins1.html#K6" title="ROBBINS1:func.6">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <font color="Maroon" title="c4">y</font></span>)</span>
<b>by </b><i><a class="ref" href="robbins1.html#D4" title="ROBBINS1:def.4">ROBBINS1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="robbins1.html#K6" title="ROBBINS1:func.6">*'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="robbins1.html#T32" title="ROBBINS1:th.32">ROBBINS1:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <font color="Maroon" title="c3">x</font>
<b>by </b><i><a class="txt" href="#E3:66_2_1_2"><i><font color="Green" title="E62">A3</font></i></a>, <a class="ref" href="robbins1.html#T15" title="ROBBINS1:th.15">ROBBINS1:15</a></i>
<br/>.= 
<font color="Maroon" title="c4">y</font> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <font color="Maroon" title="c3">x</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D5" title="LATTICES:def.5">LATTICES:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E5:66_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#NK4" title="ROBBINS1:NK.4">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#NK4" title="ROBBINS1:NK.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="robbins1.html#NK4" title="ROBBINS1:NK.4">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> <a href="robbins1.html#NK4" title="ROBBINS1:NK.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="robbins1.html#NK4" title="ROBBINS1:NK.4">+</a> <font color="Maroon" title="c3">x</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:66_2_1"/>
( <font color="Maroon" title="c1">L</font> is <a href="robbins2.html#V2" title="ROBBINS2:attr.2">satisfying_MD_1</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="robbins2.html#V3" title="ROBBINS2:attr.3">satisfying_MD_2</a> )
 <b>by </b><i><a class="txt" href="#E3:66_2_1"><i><font color="Green" title="E60">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
